Problem: Simplify and expand the following expression: $ \dfrac{2z + 8}{z - 7}+\dfrac{z}{3z + 6} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(z - 7)(3z + 6)$ Multiply the first term by $\dfrac{3z + 6}{3z + 6}$ $ \begin{align*} \dfrac{2z + 8}{z - 7} \times \dfrac{3z + 6}{3z + 6} & = \dfrac{(2z + 8)(3z + 6)}{(z - 7)(3z + 6)} \\ & = \dfrac{6z^2 + 36z + 48}{(z - 7)(3z + 6)}\end{align*} $ Multiply the second term by $\dfrac{z - 7}{z - 7}$ $ \begin{align*} \dfrac{z}{3z + 6} \times \dfrac{z - 7}{z - 7} & = \dfrac{(z)(z - 7)}{(3z + 6)(z - 7)} \\ & = \dfrac{z^2 - 7z}{(3z + 6)(z - 7)}\end{align*} $ Now we have: $ = \dfrac{6z^2 + 36z + 48}{(z - 7)(3z + 6)} + \dfrac{z^2 - 7z}{(3z + 6)(z - 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{6z^2 + 36z + 48 + z^2 - 7z}{(z - 7)(3z + 6)} $ $ = \dfrac{7z^2 + 29z + 48}{(z - 7)(3z + 6)}$ Expand the denominator: $ = \dfrac{7z^2 + 29z + 48}{3z^2 - 15z - 42}$